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I an studying about electrochemical cells, and came across the Calomel electrode. It turns out that the half cell reactions and the overall reaction are as follows.

Anode half-cell: $\text{Hg}_{2}\text{Cl}_{2}\rightarrow \text{Hg}_{2}^{2+}+2\text{Cl}^{-}$

Cathode half-cell: $\text{Hg}_{2}^{2+}+2e^{-}\rightarrow 2\text{Hg}$

Overall cell reaction: $\text{Hg}_{2}\text{Cl}_{2}+2e^{-}\rightarrow 2\text{Hg}+2\text{Cl}^{-}$

I am not sure why while writing the overall reaction, the electron was not gotten rid of. Besides why is there no net exchange of an electron in the anode half-cell reaction. Clarification would be appreciated. Thanks.

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The calomel electrode is not a cell. It can be part of a cell. It is a half-cell. A cell is made of two electrodes, one calomel electrode and another one. The reaction occurring in the calomel electrode is the reaction you present as "overall cell reaction". What you described as "anode half-cell" has nothing to do with electrochemistry. It is the equation describing how $\ce{Hg_2Cl_2}$ gets dissolved in water.

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